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mathematics
Article
A Closed Form for Slant Submanifolds of
Generalized Sasakian Space Forms
Pablo Alegre 1,*, Joaquín Barrera 2and Alfonso Carriazo 2,†
1Departamento de Economía, Métodos Cuantitativos e Historia Económica. Área de Estadística e
Investigación Operativa, Universidad Pablo de Olavide. Ctra. de Utrera, km. 1. 41013 Sevilla, Spain
2
Department of Geometry and Topology, Faculty of Mathematics, University of Sevilla, Apdo. Correos 1160,
41080 Sevilla, Spain; barreralopezjoaquin@gmail.com (J.B.); carriazo@us.es (A.C.)
*Correspondence: psalerue@upo.es
† First and third authors are partially supported by the PAIDI group FQM-327 (Junta de Andalucía, Spain)
and the MEC-FEDER grant MTM2011-22621. The third author is member of IMUS (Instituto de Matemáticas
de laUniversidda de Sevilla).
Received: 4 November 2019; Accepted: 9 December 2019; Published: 13 December 2019
Abstract:
The Maslov form is a closed form for a Lagrangian submanifold of
Cm
, and it is a conformal
form if and only if
M
satisfies the equality case of a natural inequality between the norm of the
mean curvature and the scalar curvature, and it happens if and only if the second fundamental form
satisfies a certain relation. In a previous paper we presented a natural inequality between the norm
of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space
forms, characterizing the equality case by certain expression of the second fundamental form. In this
paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form,
similar to the Maslov form, that is always closed. And, in the equality case, we studied under which
circumstances the given closed form is also conformal.
Keywords:
slant submanifolds; generalized Sasakian space forms; closed form; conformal form;
Maslov form
1. Introduction
It was proven by V. Borrelli, B.-Y. Chen and J. M. Morvan [
1
], and independently by A. Ros and F.
Urbano [
2
], that if
M
is a Lagrangian submanifold, with
dim(M) = m
, of
Cm
, with mean curvature
vector
H
and scalar curvature
τ
, then
kHk2≥2(m+2)
m2(m−1)τ
, with equality if and only if
M
is either
totally geodesic or a (piece of a) Whitney sphere. Moreover, they proved that
M
satisfies the equality
case at every point if and only if its second fundamental form σis given by
σ(X,Y) = m
m+2{g(X,Y)H+g(JX,H)JY +g(JY,H)JX}, (1)
for any tangent vector fields
X
and
Y
. Thus, they found a simple relationship between one of the main
intrinsic invariants, τ, and the main extrinsic invariant H.
It was also proven in [
2
], that the Maslov form, which is a closed form for a Lagrangian
submanifold of Cm, is a conformal form if and only if Msatisfies (1).
Later, D. E. Blair and A. Carriazo [
3
] established an analogue inequality for anti-invariant
submanifolds in
R2m+1
with its standard Sasakian structure and characterized the equality case with a
specific expression of the second fundamental form, similar to Equation (1). In a previous paper [
4
],
we studied the corresponding inequality for slant submanifolds of generalized Sasakian space forms;
Mathematics 2019,7, 1238; doi:10.3390/math7121238 www.mdpi.com/journal/mathematics
Mathematics 2019,7, 1238 2 of 15
we also characterized the equality case with an specific expression of the second fundamental form;
and finally, we presented some examples satisfying the equality case.
Both B.-Y. Chen, [
5
] and A. Carriazo, [
6
], have studied the existence of closed forms for slant
submanifolds in different environments. The existence of closed forms is particularly interesting, as
they provide conditions about submanifolds admitting an immersion in a certain environment.
The purpose of this paper was to obtain some results similar to those of [
2
] for slant submanifolds
of a generalized Sasakian space form. After a section with the main preliminaries, we show that
for a slant submanifold of a generalized Sasakian manifold, the Maslov form is not always closed.
Therefore, in the following section, we present a form that is always closed for a slant submanifold,
so it really plays the role of the Maslov form in the cited papers. Later, if the submanifold satisfies the
equality case in the corresponding inequality, that is, if the second fundamental form takes a particular
expression [4], we study if the vector field associated with the given form is a conformal vector field.
2. Preliminaries
Given a Riemannian manifold
(e
M
,
g)
, a tangent vector field
X
on
e
M
is called closed if its dual
1-form is closed. That is equivalent to
g(Y,e
∇ZX) = g(Z,e
∇YX), (2)
for all Yand Zon e
M, where e
∇is the Levi–Civita connection.
Moreover,
X
is called conformal if
LXg=ρg
, for
ρ
a function on
e
M
, where
L
is the Lie derivative.
A closed vector field Xis conformal in and only if
e
∇YX=f Y, (3)
for any tangent vector field Yon e
Mand for certain function fon e
M.
In such a case, considering an orthonormal basis
{e1
,
. . .
,
em}
on
e
M
, it holds that
e
∇eiX=f ei
,
for i=1, . . . , m.
Now, we will recall some notions about almost-contact Riemannian geometry. For more details
about this subject, we recommend the book [7].
An odd-dimensional Riemannian manifold
(e
M
,
g)
is said to be an almost contact metric manifold if
there exists on
e
M
, a
(
1, 1
)
tensor field
φ
, a unit vector field
ξ
(called the structure or Reeb vector field) and
a 1-form η, such that
η(ξ) = 1, φ2(X) = −X+η(X)ξ
and
g(φX,φY) = g(X,Y)−η(X)η(Y),
for any vector fields Xand Yon e
M. In particular, in an almost contact metric manifold we also have
φξ =0, η◦φ=0 and η(X) = g(X,ξ).
Such a manifold is said to be a contact metric manifold if
dη=Φ
, where
Φ(X
,
Y) = g(X
,
φY)
is
called the fundamental 2-form of
e
M
. The almost contact metric structure of
M
is said to be normal if
[φ
,
φ](X
,
Y) = −
2
dη(X
,
Y)ξ
, for any
X
and
Y
. A normal contact metric manifold is called a Sasakian
manifold. It can be proven that an almost contact metric manifold is Sasakian if an only if
(e
∇Xφ)Y=g(X,Y)ξ−η(Y)X,
for any Xand Yon M.
Mathematics 2019,7, 1238 3 of 15
In [
8
], J.A. Oubiña introduced the notion of a trans-Sasakian manifold. An almost contact metric
manifold e
Mis a trans-Sasakian manifold if there exists two functions αand βon e
Msuch that
(e
∇Xφ)Y=α(g(X,Y)ξ−η(Y)X) + β(g(φX,Y)ξ−η(Y)φX), (4)
for any
X
and
Y
on
e
M
. If
β=
0,
e
M
is said to be an
α
-Sasakian manifold. Sasakian manifolds appear
as examples of
α
-Sasakian manifolds, with
α=
1. If
α=
0,
e
M
is said to be a
β
-Kenmotsu manifold.
Kenmotsu manifolds are particular examples with
β=
1. If both
α
and
β
vanish, then
e
M
is a
cosymplectic manifold. In particular, from (4) it is easy to see that the following equation holds for a
trans-Sasakian manifold:
e
∇Xξ=−αφX+β(X−η(X)ξ). (5)
It was proven by J.C. Marrero that, for dimensions greater or equal than 5, the only existing
trans-Sasakian manifolds are α-Sasakian and β-Kenmotsu ones [9].
In [
10
], P. Alegre, D.E. Blair and A. Carriazo introduced the notion of a generalized Sasakian space
form as an almost contact metric manifold (e
M,φ,ξ,η,g)whose curvature tensor is given by
e
R(X,Y)Z=f1{g(Y,Z)X−g(X,Z)Y}
+f2{g(X,φZ)φY−g(Y,φZ)φX+2g(X,φY)φZ}
+f3{η(X)η(Z)Y−η(Y)η(Z)X+g(X,Z)η(Y)ξ−g(Y,Z)η(X)ξ},
(6)
where
f1
,
f2
and
f3
are differential functions on
e
M
. These manifolds are denoted by
e
M(f1
,
f2
,
f3)
; generalize the notion of Sasakian space form,
e
M(c)
, whose curvature tensor satisfies
the expression (6), with
f1=c+3
4,f2=f3=c−1
4,
where cis the constant φ-sectional curvature.
Now we recall some general definitions and facts about submanifolds. Let
M
be a submanifold
isometrically immersed in a Riemannian manifold (
e
M
,g). We denote by
∇
the induced Levi–Civita
connection on M. Thus, the Gauss and Weingarten formulas are respectively given by
e
∇XY=∇XY+σ(X,Y),
e
∇XV=−AVX+DXV,
for vector fields
X
and
Y
tangent to
M
and a vector field
V
normal to
M
, where
σ
denotes the
second fundamental form,
AV
the shape operator in the direction of
V
and
D
the normal connection.
The second fundamental form and the shape operator are related by
g(AVX,Y) = g(σ(X,Y),V). (7)
Mis called a totally geodesic submanifold if σvanishes identically.
We denote by
R
and
e
R
, the curvature tensors of
M
and
e
M
, respectively. They are related by Gauss
and Codazzi’s equations
e
R(X,Y;Z,W) = R(X,Y;Z,W)
+g(σ(X,Z),σ(Y,W)) −g(σ(X,W),σ(Y,Z)),(8)
(e
R(X,Y)Z)⊥= ( e
∇Xσ)(Y,Z)−(e
∇Yσ)(X,Z), (9)
where (e
R(X,Y)Z)⊥denotes the normal component of e
R(X,Y)Zand
(e
∇Xσ)(Y,Z) = DX(σ(Y,Z)) −σ(∇XY,Z)−σ(Y,∇XZ),
Mathematics 2019,7, 1238 4 of 15
is the derivative of Van der Waerden-Bortolotti.
On the other hand, the mean curvature vector H is defined by
H= (1/dimM)trace σ, (10)
and Mis said to be minimal if Hvanishes identically.
The scalar curvature τof Mat p∈Mis defined by
τ=∑
1≤i<j≤dimM
K(ei,ej), (11)
where
K(ei
,
ej)
denotes the sectional curvature of
M
associated with the plane section spanned by
ei
and ej, for any tangent vector fields eiand ejin a local orthonormal frame of M.
For a submanifold of an almost contact manifold, we denote
φX=TX +N X and φV=tV +nV
the tangent and normal part of
φX
and
φV
for any
X
tangent vector field and
V
normal vector field.
If the ambient space is trans-Sasakian, taking the tangent and normal part at (4) we obtain:
(∇XT)Y−tσ(X,Y)−ANY X=α(g(X,Y)ξ−η(Y)X)
+β(g(TX,Y)ξ−η(Y)TX),(12)
(∇XN)Y+σ(X,TY)−nσ(X,Y) = −βη(Y)NX, (13)
(∇Xt)V−AnV X+TAVX=βg(NX,V)ξ, (14)
(∇Xn)V+σ(X,tV) + NAVX=0. (15)
And from (5):
∇Xξ=−αTX +β(X−η(X)ξ), (16)
σ(X,ξ) = −αNX. (17)
Now, we recall the definition of slant submanifolds. These submanifolds were defined by B.-Y.
Chen in [
5
] on almost Hermitian geometry. Later, A. Lotta defined slant submanifolds on the almost
contact metric setting in [
11
]: given a submanifold
M
tangent to
ξ
, for each nonzero vector
X
tangent
to
M
at
p
, such that
X
is not proportional to
ξp
, we denote by
θ(X)
as the angle between
φX
and
TpM
.
Then,
M
is said to be slant if the angle
θ(X)
is a constant, which is independent of the choice of
p∈M
and
X∈TpM−<ξp>
. The angle
θ
of a slant immersion is called the slant angle of the immersion.
Invariant and anti-invariant immersions are slant immersions with slant angles
θ=
0 and
θ=π/
2,
respectively. A slant immersion, which is neither invariant nor anti-invariant, is called a proper slant
immersion. Slant submanifolds of Sasakian manifolds were studied by J.L. Cabrerizo, A. Carriazo,
L.M. Fernández and M. Fernández in [12,13].
From now on, we denote by
m+
1
=
2
n+
1 the dimension of
M
and 2
m+
1
=
4
n+
1 the
dimension of e
M. We assume m≥2. Then, for a slant submanifold holds:
T2X=cos2θ(−X+η(X)ξ), (18)
tN X =sin2θ(−X+η(X)ξ), (19)
NT X +nNX =0, (20)
and because of the dimensions,
n2V=−cos2θV,NtV =−sin2θVand TtV +tnV =0,
Mathematics 2019,7, 1238 5 of 15
for any X,Ytangent vector fields and Vnormal vector field.
Given a proper slant submanifold
M2n+1
, with slant angle
θ
, immersed in an almost contact
manifold
e
M4n+1
, we considered an adapted slant reference, [
6
]; it was built as follows. Given
e1
a unit
tangent vector field, orthogonal to ξ, we took:
e2= (sec θ)Te1,e1∗= (csc θ)Ne1,e2∗= (csc θ)Ne2.
For
k>
1, then proceeding by induction, for each
l=
1,
. . .
,
n−
1, we chose a unit tangent vector
field e2l+1of M, such as e2l+1, which is orthogonal to e1,e2, . . . , e2l−1,e2l,ξand took:
e2l+2= (sec θ)Te2l+1,e(2l+1)∗= (csc θ)Ne2l+1,e(2l+2)∗= (csc θ)Ne2l+2.
In this way
{e1, . . . , em,ξ,e1∗, . . . , em∗}(21)
is an orthonormal reference such that
e1
,
. . .
,
em
belong to the contact distribution,
D
and
e1∗
,
. . .
,
em∗
are normal to M. Moreover, it can be directly computed that:
Te2j−1= (cos θ)e2j,Te2j=−(cos θ)e2j−1,j=1, . . . , k;
Nei= (sin θ)ei∗,tei∗=−(sin θ)ei,i=1, . . . , m;
ne(2j−1)∗=−(cos θ)e(2j)∗,ne(2j)∗= (cos θ)e(2j−1)∗,j=1, . . . , k.
Finally, a slant submanifold of an
(α
,
β)
trans-Sasakian generalized Sasakian space form
e
M2m+1(f1
,
f2
,
f3)
, is called
∗
-slant submanifold, [
4
], if its second fundamental form
σ
is given by the
following expression:
σ(X,Y) = m+1
m+2n(g(X,Y)−η(X)η(Y))H
+1
sin2θg(φX,H)−αm+2
m+1η(X)NY
+1
sin2θg(φY,H)−αm+2
m+1η(Y)NX.
(22)
They are specially interesting because it was proven in [
4
] that this expression of the second
fundamental form characterizes the equality case of the following inequality involving the squared
mean curvature kHk2and the scalar curvature τ:
(m+1)2kHk2−2m+2
m−1τ≥ − m(m+2)
m−1((m+1)f1+3f2cos2θ−2f3−2αsin2θ). (23)
3. The Maslov Form
For any submanifold of any almost contact manifold, we consider the Maslov form
ωH
as the
dual form of φH; that is
ωH(X) = g(X,φH),
for any Xtangent vector field in the submanifold. We can also define a canonical 1-form on Mby
Θ=
m
∑
1=1
ωi∗
i,
where ωi∗
iare the connection forms given by Cartan’s structure equations.
Mathematics 2019,7, 1238 6 of 15
We can relate these two forms for certain slant submanifolds. In [
12
], proper slant submanifolds
such as for any tangent vector fields Xand Ywere studied with:
(∇XT)Y=cos2θ(g(X,Y)ξ−η(Y)X). (24)
They were called slant Sasakian submanifolds in [
6
]; however, we can point that they are
α
-Sasakian
manifolds with the induced structure
φ=sec θT
. That aims us to defined slant trans-Sasakian
submanifolds as those verifying:
(∇Xφ)Y=α(g(X,Y)ξ−η(Y)X) + β(g(φX,Y)ξ−η(Y)φX). (25)
For a slant trans-Sasakian submanifold of a trans-Sasakian manifold the relation between the
structure functions is given by
sec θα =αand β=β. (26)
From (25) and (12) it is deduced that
ANY X=ANXY+αsin2θ(η(Y)X−η(X)Y), (27)
for any X,Ytangent vector fields.
Then, for such a submanifold, the relation between
Θ
and the Maslov form is given in the
following theorem.
Theorem 1.
Let
Mm+1
be a slant trans-Sasakian submanifold of a generalized Sasakian space form
e
M2m+1(f1,f2,f3)endowed with an (α,β)trans-Sasakian structure. Then:
ωH=−sin θ
m+1(Θ+mαsin θη). (28)
Proof. Considering an adapted slant basis, it holds
ωH(ei) = g(ei,φH) = −g(Nei,H) = −sin θg(ei∗,H), (29)
for i=1, . . . , m. Moreover,
Θ=
2n
∑
l=1
2n
∑
i=1
σl∗
li ωi+
2n
∑
l=1
σl∗
lξη. (30)
But,
σl∗
lξ=g(σ(el,ξ),el∗) = −csc θg(Nel,N el) = −sin θ, (31)
and
σl∗
li =g(σ(el,ei),el∗) = csc θg(σ(el,ei),Nel)
=csc θg(ANelei,el) = csc θg(AN eiel,el)
=g(σ(el,el),ei∗) = σi∗
ll ,
(32)
where we have used (27); that is, Mis a slant trans-Sasakian submanifold.
And therefore, from (30)–(32),
Θ+mαsin θη =∑
i=1
2n(trσi∗)ωi.
Mathematics 2019,7, 1238 7 of 15
As σ(ξ,ξ) = 0:
H=1
m+1
m
∑
j=1
σ(ej,ej). (33)
Now, from (29) and (33), it holds that
ωH(ei) = −sin θ
m+1
2k
∑
j=1
σi∗
jj =−sin θ
m+1(Θ+mαsin θη)(ei),
for i=1, . . . , m. Finally, as ωH(ξ) = g(tH,ξ) = 0, the proof is finished.
Following the same steps that [
5
] did for slant submanifolds of an almost Hermitian manifold or
[
6
] for an almost contact manifold, and after a long computation, the differentials of
θ
and
η
can be
proven. The proof is straightforward so we have omitted it.
Lemma 1.
Let
Mm+1
, a proper slant submanifold of a generalized Sasakian space, form
e
M2m+1
endowed with
an (α,β)trans-Sasakian structure, with M tangent to ξand m ≥2. Then, the 1-forms Θand ηsatisfy:
dΘ=−2 sin θcos θ(α2+f2(m+1)) k
∑
j=1
ω2j−1∧ω2j−
k
∑
j=1
ω(2j−1)∗∧ω(2j)∗!
+(−2 sin2θ(α2+f2(m+1))+α2+f2−f1−β2)
k
∑
j=1
ω2j−1∧ω(2j−1)∗+
k
∑
j=1
ω2j∧ω(2j)∗!,
(34)
and
dη=−2αcos θ
k
∑
j=1
ω2j−1∧ω2j−2αsin θ
k
∑
j=1
ω2j−1∧ω(2j−1)∗−
−2αsin θ
k
∑
j=1
ω2j∧ω(2j)∗+2αcos θ
k
∑
j=1
ω(2j−1)∗∧ω(2j)∗,
(35)
where θis the slant angle of M.
As we are considering a trans-Sasakian manifold with a dimension greater or equal than 5,
from [
9
], it must be an
α
-Sasakian or a
β
-Kenmotsu manifold. So we distinguish both two cases in the
following theorems.
Theorem 2.
Let
Mm+1
be a proper slant trans-Sasakian submanifold of a connected generalized Sasakian space
form
e
M2m+1(f1
,
f2
,
f3)
endowed with an
α
-Sasakian structure. Then, the Maslov form is closed if and only if
f1=0. In such a case, it holds f2=f3=−α2.
Proof. As e
M2m+1is α-Sasakian, from Proposition 4.1 of [14], αis constant. From (28),
dωH=−sin θ
m+1(dΘ+mαsin θdη).
Then, from (34) and (35), it is deduced that d
ωH=
0 if and only if it holds
α2+f2=
0 and
f1=
0.
Moreover, Theorem 4.2 of [
14
] establishes that both conditions are equivalent, as
f1−α2=f2=f3
.
Remark 1.
If the ambient space is a Sasakian space form
e
M2m+1(c)
, the Maslov form is closed if and only if
c=−3, as it was proved in [6].
Mathematics 2019,7, 1238 8 of 15
Theorem 3.
Let
Mm+1
be a proper slant trans-Sasakian submanifold of a generalized Sasakian space form
e
M2m+1(f1,f2,f3)endowed with a β-Kenmotsu structure. Then, the Maslov form is closed if and only if
f1=−β2and f2=0.
In such a case, it holds f3=ξ(β).
Proof.
Again from (28), (34) and (35), d
ωH=
0 if and only if
f2=
0 and
f1+β2=
0. The last condition
is obtained from Proposition 4.3 in [14], where it was proven that f1−f3+ξ(β) + β2=0.
Remark 2.
We note that on the opposite that for Lagrangian submanifold of
Cn
, [
2
], or totally real submanifolds
of
R2m+1
, [
3
], the Maslov is not always closed. That aims us to look for an adapted form that is closed in
more cases.
4. An Adapted Closed Form
As the Maslov form is not always closed for slant submanifolds it is necessary to find a new form
related with this Maslov form but including the special slant character of the submanifold.
Both the Maslov form and
Θ
can be considered forms at
e
M
or
M
. As both
η
and
Θ
vanish at
TM⊥
, it is the same defining them on
e
M
or
M
; however, it is not the same considering
dη
or
dηeM
and
dΘ
or
dΘeM
. Although both B.-Y. Chen and A. Carriazo, [
5
] and [
6
], studied conditions for
dωH
and
dΘ
vanishing at the manifold; their real interest was finding a closed form at the submanifold, not at
the manifold.
Therefore, we consider the restrictions of
Θ
and
η
at the submanifold. From (34) and (35) it
is deduced:
dηeM=−2αcos θ
m
∑
j=1
ω2j−1∧ω2j(36)
and
dΘeM=−2 sin θcos θ(α2+f2(m+1))
m
∑
j=1
ω2j−1∧ω2j. (37)
So we find that, for obtaining a closed form, the relation between
Θ
and
η
is not the given by the
Maslov form at (28).
Again, we particularize to
α
-Sasakian or a
β
-Kenmotsu manifolds. Firstly, we consider an
α
-Sasakian manifold. It was proven in [
14
], that if
α6=
0 and
e
M(f1
,
f2
,
f3)
is connected, then
α
is
constant, and the functions are constant and related by f1−α2=f2=f3. We can write:
f1=c+3α2
4,f2=f3=c−α2
4.
From now on, we suppose e
Mis connected.
Lemma 2.
Let
Mm+1
be a slant submanifold of an
α
-Sasakian generalized Sasakian space form
e
M2m+1(f1,f2,f3), with α6=0. Then, the form Θ−sin θα2+f2(m+1)
αηis closed at M.
Proof.
It is directly deduced from (36) and (37) that
αdΘ−sin θ(α2+f2(m+
1
))dη=
0, and as
α
is
constant, the result is proven.
Moreover, the field associated to the closed form is
−m+1
sin θtH −sin θm+α2+f2(m+1)
αξ, (38)
Mathematics 2019,7, 1238 9 of 15
so we already have the following theorem.
Theorem 4.
Let
Mm+1
be a slant submanifold of an
α
-Sasakian generalized Sasakian space form
e
M2m+1(f1,f2,f3), with α6=0. Then, the field tH +sin2θ
m+1
mα+α2+f2(m+1)
αξis closed.
Corollary 1.
Let
Mm+1
a slant submanifold of a Sasakian space form
e
M2m+1(c)
; the field
tH +sin2θc+3
4ξ
is closed.
Note that this result improves the one obtained by A. Carriazo in [
6
] giving a closed form for a
slant submanifold of any Sasakian space form.
Corollary 2.
Let
M2m+1
be a compact and simply connected manifold. Then,
M
can not be immersed in a
generalized Sasakian space form,
e
M4m+1(
0,
−α2
,
−α2)
, endowed with an
α
-Sasakian structure,
α6=
0, like a
slant submanifold.
Proof.
If
Mm+1
is a slant submanifold of
e
M4m+1(
0,
−α2
,
−α2)
, with an
α
-Sasakian structure.
By Theorem 4the vector field
tH +sin2θ
m+1
mα+α2+f2(m+1)
αξ6=0,
is closed, and the corresponding form is also closed. Therefore it represents a cohomology class in
H1(M
;
R)
. But, as
M
is compact, it can not be an exact form. So
H1(M
;
R)
is a nontrivial cohomology
class and Mcould not be simply connected what is a contradiction.
On the other hand, for a
β
-Kenmotsu manifold
dη=
0 and from Theorem 1,
ωH=−sin θ
m+1Θ
.
The following lemma studies when it is a closed form.
Lemma 3.
Let
Mm+1
be a proper slant submanifold of a
β
-Kenmotsu generalized Sasakian space form
e
M2m+1
,
with M tangent to ξand m ≥2. Then, the Maslov form at M is closed if and only if f2=0.
Proof. For a β-Kenmotsu manifold ωH=−sin θ
m+1Θ. And writing (37) for α=0,
dΘeM=−2 sin θcos θf2(m+1)
m
∑
j=1
ω2j−1∧ω2j. (39)
Therefore, the Maslov form is closed in Mif and only f2=0.
Note, that in such a case
f1−f3+ξ(β) + β2=
0 ([
14
], Proposition 4.3). Moreover, we observe that,
on the opposite that for
α
-Sasakian manifolds, we cannot find a closed form for a slant submanifold of
any generalized Sasakian space form with a β-Kenmotsu structure.
However, for f2=0, we have obtained a closed vector field as follows.
Theorem 5.
Let
Mm+1
be a slant submanifold of an
β
-Kenmotsu generalized Sasakian space form
e
M2m+1(f1, 0, f3). Then, the field tH +sin2θm
m+1ξis closed.
Again, we can present a topological obstruction for slant immersions:
Mathematics 2019,7, 1238 10 of 15
Corollary 3.
Let
M2m+1
be a compact and simply connected manifold. Then,
M
cannot be immersed in
a generalized Sasakian space form,
e
M4m+1(f1
, 0,
f3)
, endowed with an
β
-Kenmotsu structure, as a slant
submanifold.
From now on, we will write tH +aξand tH +bξ, with
a=sin2θ
m+1
mα+α2+f2(m+1)
αand b=sin2θm
m+1,
for the correspondent closed vector fields.
5. About Conformal Forms for α-Sasakian Space Forms
As we said in the Introduction, for those Lagrangian submanifolds of
Cm
verifying the equality
case, the Maslov form, that is closed, is also conformal. Now we study if the closed form presented in
the previous section is conformal for those slant submanifolds verifying the equality case at (23).
We are considering a connected manifold, so α,f1,f2and f3are constant functions.
We want to compute
∇X(tH +aξ)
, for any
X
tangent vector field. It is a long computation.
Firstly, we compute
∇N
for later use. Using the expression of the second fundamental form of a *-slant
submanifold, (22), and (20) in (13):
(∇XN)Y=m+1
m+2n(g(X,Y)−η(X)η(Y))nH −g(X,TY)H
+21
sin2θg(φX,H)−m+2
m+1η(X)nNY
+1
sin2θg(φY,H)−m+2
m+1η(Y)nN X
−1
sin2θg(φTY,H)N X.
(40)
Lemma 4.
Let
M
be *-slant submanifold of an generalized Sasakian space form
e
M(f1
,
f2
,
f3)
endowed with an
α-Sasakian structure. For every X tangent vector field belonging to the contact distribution it holds:
∇X(tH +aξ) = −g(DXH,NX)X+1
sin2θ
m+1
m+2g2(H,NX)−aTX
−3
sin2θ
m+1
m+2g(H,NX)tnH
+1
sin2θ
m+1
m+2g(H,nN X)tH +g(H,nNX)ξ.
(41)
Proof.
Firstly, from Codazzi’s equation we will compute
DXH
, and after,
∇X(tH +aξ)
.
Writing Codazzi’s equation, (9), for a generalized Sasakian space form, for any unit orthogonal
X,Ytangent vector fields in the contact distribution, using (3) (6), R(X,Y)Y)⊥gives:
3m+2
m+1f2g(X,TY)NY =DXH+3m+2
m+1g(Y,TX)NY
+2
sin2θ{g((∇XN)Y,H)NY +g(NY,DXH)NY +g(NY,H)(∇XN)Y}
+1
sin2θ{−g((∇YN)X,H)NY −g((∇YN)Y,H)NX −g(NX,H)(∇YN)Y
−g(NX,DYH)NY −g(NY,DYH)NX −g(NY,H)(∇YN)X}.
(42)
Mathematics 2019,7, 1238 11 of 15
Then, using (40), we obtain:
3m+2
m+1(f2+1)g(X,TY)NY =DXH
+1
sin2θ{2g(NY,DXH)NY −g(NX,DYH)NY −g(NY,DYH)NX}
+1
sin2θ
m+1
m+2−g(NX,H)nH −2
sin2θg(NY,H)g(n NY,H)NX −3g(NY,H)g(X,TY)H
−3g(X,TY)kHk2+4
sin2θg(NX,H)g(nNY,H)−2
sin2θg(NY,H)g(n NX,H)NY.
(43)
At this point, we use that, taking into account Corollary 1,tH +aξis a closed vector field.
g(∇X(tH +aξ),Y) = g(∇Y(tH +aξ),X).
Then, using (16)
g(∇XtH,Y) = g(∇YtH,X)−2ag(TY,X),
and therefore, (14) gives
g(NX,DYH) = −g(X,tDYH) = −g(X,∇YtH −AnH Y+TAHY)
=g(NY,DXH) + 2ag(TX,Y)−g(σ(TY,X),H) + g(σ(Y,TX),H).(44)
Now, using (22) carries to
g(NX,DYH) =g(NY,DXH) + 2ag(TX,Y) + m+1
m+22g(Y,TX)kHk2
+m+1
m+2
2
sin2θ(g(NY,H)g(N TX,H)−g(NX,H)g(NTY,H)).
(45)
So (43) gives
3m+2
m+1(f2+1)g(X,TY)NY =DXH+1
sin2θ{g(NY,DXH)NY −g(NY,DYH)N X}
+1
sin2θ
m+1
m+2−2am+2
m+1g(TX,Y) + g(Y,TX)kHk2+2
sin2θg(NX,H)g(nNY,H)NY
−g(NX,H)nH −2
sin2θg(NY,H)g(n NY,H)NX +3g(NY,H)g(Y,TX)H.
(46)
Now, for dimensions over or equal than 5, we can consider
X
orthogonal to
Y
and
TY
. Multiplying
by NX,
0=g(DXH,NX)−g(NY,DYH)
+1
sin2θ
m+1
m+2{−g(N X,H)g(nNX,H)−2g(NY,H)g(nNY,H)}.(47)
Interchanging Xand Yat (47), and adding it to the previous equation:
g(NX,H)g(nN X,H) = −g(NY,H)g(nNY,H). (48)
For TY, that is also orthogonal to X,TX,
g(NX,H)g(nN X,H) = −g(N TY,H)g(nN TY,H) = cos2θg(NY,H)g(n NY,H). (49)
Mathematics 2019,7, 1238 12 of 15
From (48) and (51), we get
g(NX
,
H)g(nN X
,
H) =
0 for every
X
unit vector field in the contact
distribution. Moreover, developing 0 =g(N(X+Y),H)g(nN(X+Y),H), we obtain
g(NX,H)g(nNY,H) = −g(NY,H)g(nN X,H). (50)
Also, at (47), we get
g(DXH,NX) = g(NY,DYH), (51)
so g(DXH,NX)is independent of Xunit vector field in the contact distribution.
Now, multiplying (46) by NY,
0=2g(DXH,NY) + 3
sin2θ
m+1
m+2g(NX,H)g(nNY,H). (52)
But (44) for any X, a unitary vector field orthogonal to Y,TY, in the contact distribution it states:
g(NX,DYH)−g(σ(TX,Y),H) = g(NY,DXH)−g(σ(TY,X),H). (53)
Using (52) and (22) at (53)
−7
2 sin2θ
m+1
m+2g(NX,H)g(nNY,H) = −7
2 sin2θ
m+1
m+2g(NY,H)g(n NX,H), (54)
where X,Ycan be interchanged. Comparing (50) with (54) it is proven that
g(NX,H)g(nNY,H) = 0, (55)
and consequently, by (52),
g(DXH,NY) = 0, (56)
for each Xorthogonal to Yand TY at the contact distribution.
It only rests on us to compute
g(DXH
,
NT X)
in order to know
DXH
. Multiplying (46) by
NT X
we obtain:
g(DXH,NT X) = −cos2θ
sin2θ
m+1
m+2g(H,NX)2. (57)
Therefore, taking an orthogonal basis {e∗
1, ..., e∗
n}at T⊥M,
DXH=∑g(DXH,e∗
j)e∗
j=
=1
sin2θg(DXH,NX)NX −1
sin4θ
m+1
m+2g(H,NX)2NTX,(58)
for any Xunit tangent field orthogonal to ξ.
Finally, for any Xat the contact distribution, and any Ztangent vector field,
g(∇X(tH +aξ),Z) = g(∇XtH,Z)−ag(TX,Z)
=g(t∇XH+AnH X−TAHX,Z)−ag(TX,Z)
=−g(DXH,NZ) + g(nH,h(X,Z)) + g(H,h(X,TZ)) −ag(T X,Z)
=−gg(DXH,NX)NX −1
sin4θ
m+1
m+2g(H,NX)2NTX,NZ
+m+1
m+21
sin2θg(NX,H)g(NZ,nH) + 1
sin2θg(NZ,H)−m+2
m+1η(Y)g(NX,nH)
Mathematics 2019,7, 1238 13 of 15
+m+1
m+21
sin2θg(NX,H)g(NTZ,H) + 1
sin2θg(NT Z,H)g(N X,H)
−ag(T X,Z), (59)
where we used (22). This last equation, using (19), direct gives the desired expression of
∇XtH +
c−1
4cos2θξ.
The quid point of the above proof is to deduce, from Codazzi’s equation and the expression of the
second fundamental form, that
g(DXH
,
NX)
is independent of
X
and also that
g(DXH
,
NY) =
0 for
Y
orthogonal to X,TX. This is the same sketch than A. Ros and F. Urbano did in [2].
Now, we repeat the same steps in order to obtain ∇ξ(tH +aξ).
Lemma 5. Let M be *-slant submanifold of a Sasakian space form e
M(c); it holds:
∇ξ(tH +aξ) = −TtH. (60)
Proof. Using that, from Corollary 1,tH +aξis a closed vector field,
g(∇ξ(tH +aξ),Y) = g(∇Y(tH +aξ),ξ),
so using (16),
g(∇ξtH,X) = g(∇XtH,ξ) = −g(tH,∇Xξ) = g(tH,T X) = −g(TtH,X),
for any Xtangent vector field, which finishes the proof.
Theorem 6.
Let
M
be *-slant submanifold of a generalized Sasakian space form
e
M(f1
,
f2
,
f3)
, endowed with
an α-Sasakian structure. Then, for every X tangent vector field it holds:
∇X(tH +aξ) = (−g(DXH,NX) + η(X)g(H,nNX))(X−η(X)ξ)
+1
sin2θ
m+1
m+2g(H,NX)2−aTX
+−3
sin2θ
m+1
m+2g(H,NX) + η(X)tnH
+1
sin2θ
m+1
m+2g(H,nN X)tH +g(H,nNX)ξ.
Proof. It is a direct consequence of Lemmas 4and 5.
So, in general, for a *-slant submanifold of a generalized Sasakian space form, the closed form is
not conformal. However, for the corresponding vector field, the covariant derivative with respect to
X
is in the direction of X,TX,tnH and ξ.
6. About Conformal Forms for β-Kenmotsu Space Forms
At Section 4we obtained that, for a
β
-Kenmotsu generalized Sasakian space form
e
M(f1
, 0,
f3)
,
the vector field
tH +sin2θm
m+1ξ=tH +bξ
is always closed. So, the associated form plays the role
of the Maslov form for Lagrangian submanifolds of Kaehler manifolds. In this section we study if it is
conformal for a *-slant submanifold.
The study is similar to the one made at Section 5, so we omit the proofs.
Mathematics 2019,7, 1238 14 of 15
Lemma 6.
Let
M
be *-slant submanifold of a
β
-Kenmotsu generalized Sasakian space form
e
M(f1
, 0,
f3)
.
Then, for every X tangent vector field belonging to the contact distribution it holds:
∇X(tH +bξ) = −g(DXH,NX) + βsin2θm
m+1X
+m+1
m+21
sin2θg(H,NX)2− kHk2TX
−3
sin2θ
m+1
m+2g(H,NX)tnH
+1
sin2θ
m+1
m+2g(H,nN X)tH +βg(NX,H)ξ.
(61)
Again, the quid point of the proof is to deduce, from Codazzi’s equation and the expression of
the second fundamental form, that
g(DXH
,
NX)
is independent of
X
and that
g(DXH
,
NY) =
0 for
X
orthogonal to Y.
Now, we repeat the same steps in order to obtain ∇ξ(tH +bξ).
Lemma 7. Let M be *-slant submanifold of a β-Kenmotsu space form e
M(f1, 0, f3); it holds:
∇ξtH =−βtH. (62)
Finally, we get:
Theorem 7.
Let
M
be *-slant submanifold of a
β
Kenmotsu space form
e
M(f1
, 0,
f3)
. Then, for every
X
tangent
vector field it holds:
∇X(tH +bξ) =
=−g(DXH,NX) + βη (X)g(H,nNX) + βsin2θm
m+1(X−η(X)ξ)
+m+1
m+21
sin2θg(H,NX)2− kHk2TX −3
sin2θ
m+1
m+2g(H,NX)tnH
+1
sin2θ
m+1
m+2g(H,nN X)tH −βη (X)tH +βg(NX,H)ξ.
Proof. It is a direct consequence from Lemmas 6and 7.
Again, for a *-slant submanifold of a
β
-Kenmotsu generalized Sasakian space form, the closed
form is not conformal. However, for the corresponding vector field, the covariant derivative with
respect to Xis in the direction of X,TX,tH,tnH and ξ.
Author Contributions: All the authors contributed equally to this work.
Conflicts of Interest: The authors declare no conflict of interest.
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